\documentclass[reqno]{amsart}
\usepackage{hyperref}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2015 (2015), No. 121, pp. 1--9.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2015 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2015/121\hfil Nonlinear damped Schr\"odinger equation]
{Nonlinear damped Schr\"odinger equation in two space dimensions}

\author[T. Saanouni \hfil EJDE-2015/121\hfilneg]
{Tarek Saanouni}

\address{Tarek Saanouni \newline
University Tunis El Manar,
Faculty of Sciences of Tunis,
LR03ES04 partial differential quations and applications,
2092 Tunis, Tunisia}
\email{Tarek.saanouni@ipeiem.rnu.tn}

\thanks{Submitted November 28, 2014. Published April 30, 2015.}
\subjclass[2010]{35Q55}
\keywords{Nonlinear damped Schr\"odinger equation;
existence; uniqueness; \hfill\break\indent Moser-Trudinger inequality, decay}

\begin{abstract}
 In this article, we study the initial value problem for a 
 semi-linear damped  Schr\"odinger equation with exponential growth 
 nonlinearity  in two space dimensions.  We show global well-posedness and
 exponential decay.

\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks

\section{Introduction}

Consider the initial value problem for a damped semilinear Schr\"odinger equation
\begin{equation}\label{eq1}
\begin{gathered}
i\dot u+\Delta u-\alpha u+\omega \Delta\dot u-\mu\dot u=\epsilon f(u),\\
u|_{t=0}= u_0,\\
u|_{\partial\Omega}=0.
\end{gathered}
\end{equation}
This equation arises for instance in plasma physics \cite{nb} or 
in optical fibers models \cite{jj}. Here and hereafter 
$(\alpha,\mu,\omega)\in\mathbb{R}_+^3$ and $\epsilon\in\{\pm 1\}$. 
The set $\Omega\subset\mathbb{R}^2$ is a bounded smooth domain and 
$u(t,x):\mathbb{R}_+\times\Omega\to\mathbb{C}$. The nonlinearity $f$ satisfies the 
Hamiltonian form $f(z)=zF'(|z|^2)$, where $F\in C^1(\mathbb{R}_+)$ and vanishes on zero.
 Moreover, we assume that for all  $\alpha>0$, there exists $C_\alpha>0$ such that
\begin{equation}\label{f}
|f(z_1)-f(z_2)|^2\leq C_\alpha |z_1-z_2|^2
\big(e^{\alpha |z_1|^2}-1+e^{\alpha |z_2|^2}-1\big),\quad \forall z_1,z_2\in\mathbb{C}.
\end{equation}
We define the energy of a solution $u$ to \eqref{eq1} by
$$
E(t)=E_\alpha(u(t)):=\int_\Omega\Big(|\nabla u(t)|^2+\alpha | u(t)|^2
+\epsilon F(|u(t)|^2)\Big)\,dx.
$$
The decay of the energy formally satisfies
$$
\dot E(t)=-\omega \|\nabla \dot u\|_{L^2}^2-\mu\|\dot u\|_{L^2}^2.
$$
If $\epsilon=-1$, the energy is positive and \eqref{eq1} is said to be 
defocusing, otherwise it is focusing.

 In the monomial case $f(u)=u|u|^{p-1}$, local well-posedness in the energy 
space holds for any $1<p<\infty$ \cite{G.V,Cas.F}. Moreover, the solution 
is global if $1<p<3$ or in the defocusing case \cite{Cas}. So it is natural 
to consider problems with exponential nonlinearities, which have several 
applications, as for example the self trapped beams in plasma \cite{Lam}. 
Moreover, the Moser-Trudinger estimate \cite{Ad} provides another motivation 
to consider exponential type nonlinearity in order to study semilinear 
Schr\"odinger equation in two space dimensions.

 The two dimensional Schr\"odinger problem with exponential growth nonlinearity
 was studied in \cite{Na}, where global well-posedness and scattering were proved. 
Later on, the critical type nonlinearity was considered in \cite{Col.I}. 
In fact, global well-posedness for small data in the subcritical and critical 
cases holds. Moreover,  scattering in the subcritical case was established. 
The author \cite{T2} obtained a decay result in the critical case. 
Recently \cite{T}, global well-posedness and scattering in the energy space 
without any condition on the data, for some weaker exponential nonlinearity,
were proved (the associated wave problem was treated in \cite{OMTS,OMTS1}).

It is the aim of this article is to extend previous results about 
global well-posedness of the classical Schr\"odinger problem in two space 
dimensions with exponential type nonlinearity to the damped case.

he rest of the article is organized as follows. 
The second section states the main results and gives some tools needed in the sequel. 
The third section deals with local well-posedness of \eqref{eq1}. 
In the last section we prove global well-posedness of \eqref{eq1} in 
the focusing case and an exponential decay of the energy.

We mention that $C$ will be used to denote a constant which may vary from 
line to line.
We use $A\lesssim B$ to denote an estimate of the form $A\leq C B$
for some absolute constant $C$. We denote Lebesgue space $L^p:=L^p(\Omega)$ 
and Sobolev space $H_0^1:=H_0^1(\Omega)$ endowed with the complete norm 
$\|\cdot\|_{H_0^1}:=\|\nabla\cdot\|_{L^2}$.
 Finally, if $T>0$ and $X$ is an abstract space, we denote 
$C_T(X):=C([0,T],X)$ and $L_T^p(X):=L^p([0,T],X)$.

\section{Results and background}

In this section, we give the main results of this paper and some technical 
tools needed in the sequel. For $u\in H_0^1$, we define the quantities
\begin{gather*}
I_\alpha(u):=\int_\Omega\Big(|\nabla u|^2+\alpha |u|^2-\bar uf(u)\Big)\,dx;\\
m:=\inf_{0\neq u\in H_0^1}\{E(u),\,I(u)=0\},\quad 
N:=\{0\neq u\in H_0^1: I(u)=0\};\\
N^+:=\{u\in H_0^1: I(u)>0\}\cup\{0\};\\
(u,v)_*:=\omega(\nabla u,\nabla v)_{L^2}+\mu (u,v)_{L^2},\quad
 \|\cdot\|_*^2:=(u,u)_*.
\end{gather*}
$E_T:=C_T(H_0^1)$ endowed with the norm
$\|\cdot\|_T:=\|\cdot\|_{L_T^\infty(H_0^1)}$.
If $u=u(t)$, we denote for simplicity $I(t)=I_\alpha(u(t))$.
The first result is about the existence of a unique local solution to \eqref{eq1}.

\begin{theorem}\label{thm1}
Assume that $\mu>0$, the nonlinearity satisfies \eqref{f}, and  $u_0\in H_0^1$. 
Then there exists $T>0$ and a unique local solution to the Cauchy problem 
\eqref{eq1}, in the energy space
$$
C([0,T],H_0^1).
$$
Moreover,
\begin{enumerate}
\item
the solution satisfies decay of the energy;
\item
the solution is global in the defocusing case.
\end{enumerate}
\end{theorem}

In the next result, we assume that the nonlinearity satisfies the supplementary 
condition: There exist $r_0,a>0$ such that  
\begin{equation}\label{as}
F(r_0)>0 \quad\text{and}\quad
rf(r)\geq (1+a) F(r)\quad\text{for all }\quad r\in\mathbb{R}_+.
\end{equation}
In the focusing case, we give a result of global existence and exponential decay.

\begin{theorem}\label{thm3}
Assume that $\epsilon=-1$, $\omega>0$ and the nonlinearity satisfies \eqref{f}
 with \eqref{as}. Let $u_0\in N^+$ such that $E(0)<m$. Then the solution $u$ 
given by the previous result is global and satisfies
\begin{enumerate}
\item $u(t)\in N^+$ for any time;

\item for $0<\alpha$ large enough, there exists  $\gamma>0$ such that
\[
 0<\|u(t)\|_{H_0^1}\lesssim e^{-\gamma t}, \quad\forall t\in\mathbb{R}_+.
\]
\end{enumerate}
\end{theorem}

\begin{remark} \rm
The following function satisfies conditions of Theorem \ref{thm3},
$$
f(u):=\frac12  u(1+{|u|}^2)^{\frac{-1}{2}}
\Big({\rm e}^{(1+{|u|}^2)^{\frac{1}{2}}}-e(1+|u|^2)^{\frac{1}{2}}\Big).
$$
\end{remark}

\begin{proof}
 We have $F(r)=e^{(1+r)^{\frac12}}-\frac{e}2(r+2)=e^t-\frac{e}2(t^2+1)$, 
where $t:=\sqrt{1+r}$. From direct computations, we have
\begin{gather*}
rF'(r)=\frac12(-1+t^2)(\frac{e^t}t-e);\\
\phi_a(t):=2(rF'(r)-(1+a)F(r))=(t-\frac1{t}-2(1+a))e^t+ea(1+t^2)+2e ;\\
\phi_a'(t)=(t-\frac1{t}-1-2a+\frac1{t^2})e^t+2eat,\quad 
 \phi_a(1)=0=\phi_a'(1); \\
\phi_a''(t)=(t-\frac1{t}+\frac2{t^2}-\frac2{t^3}-2a)e^t+2ea,\quad \phi_a''(1)=0;\\
\phi_a'''(t)=(t-\frac1{t}+\frac3{t^2}-\frac6{t^3}+\frac6{t^4}+1-2a)e^t,\quad 
\phi_a'''(1)=2(2-a)e.
\end{gather*}
Now, taking $\phi_a'''(t)=(\psi(t)+1-2a)e^t$, where 
$t^4\psi(t)=t^5-t^3+3t^2-6t+6\geq 0$ for $t\geq 1$. Which implies 
that \eqref{as} is satisfied for any $a\in(0,1/2)$.
\end{proof}

In the two-dimensional space, we have the Sobolev injections \cite{AC1},
$$
H_0^1\hookrightarrow L^p,\quad\text{for any}\quad 2\leq p<\infty,
$$
and it is false for $p=\infty$.
 The critical Sobolev embedding is described with the so called Orlicz space 
\cite{HMN}, which is given by the following Moser-Trudinger inequality 
\cite{Ad,Mo,Tr}.

\begin{proposition} \label{prop2.4}
Let $\alpha\in (0,4\pi)$. Then there exists  a constant $\mathcal{C}_{\alpha}$ 
such that for all $u\in H^{1}_0$ satisfying $\|\nabla u\|_{L^{2}}\leq 1$, one has
$$
\int_{\Omega}\Big(e^{\alpha |u(x)|^2}-1\Big)\,dx\leq
\mathcal{C}_{\alpha}\|u\|^{2}_{L^{2}}.
$$
Moreover,
\begin{enumerate}
\item the above inequality is false when $\alpha\geq 4\pi$;

\item $\alpha=4\pi$ becomes admissible if we consider 
$\|u\|_{H^{1}_0}\leq 1$ rather than $\|\nabla u\|_{L^{2}}\leq 1$. 
In this case, one has
$$
\sup_{\|u\|_{H^{1}_0}\leq 1} \int_{\Omega}e^{4\pi|u(x)|^2}\,dx<\infty
$$
and this is false for $\alpha>4\pi$ \cite{Ad}.
\end{enumerate}
\end{proposition}
 
\section{Proof of Theorem \ref{thm1}}

We prove well-posedness of the Cauchy problem \eqref{eq1} in the energy space. 
We take in this section $\epsilon=1$, in fact the sign of the nonlinearity 
has no local effect.

\subsection{Local well-posedness}

\begin{lemma}\label{lem1}
Let $T>0$, $u_0\in H_0^1$ and $u\in C_T(H_0^1)$. Then there exists a unique
$v\in E_T$ such that 
\begin{equation} \label{eq2}
\begin{gathered}
i\dot v+\Delta v-\alpha v+\omega \Delta\dot v-\mu\dot v=f(u) \quad\text{on }
 [0,T]\times\Omega, \\
v|_{t=0}=u_0,\\
v|_{\partial\Omega}=0.
\end{gathered}
\end{equation}
\end{lemma}

\begin{proof}
Let  $W_h:=\langle w_1,\dots,w_h\rangle$, where $\{w_j\}$ is a complete 
system of eigenvectors of $-\Delta$ in $H_0^1$ such that $\|w_j\|_{L^2}=1$. 
Then, $\{w_j\}$ is orthogonal and complete on $L^2$ and $H_0^1$. 
Denote the associated eigenvalues $\{\lambda_j\}$. Let
$$
u_0^h:=\sum_1^h\Re\Big(\int_\Omega\overline{\nabla u_0}\nabla w_j\Big)w_j.
$$
Then, $u_0^h\in W_h$ and $u_0^h\rightarrow u_0$ in $H_0^1$. 
For $h\geq 1$, we seek for $h$ functions $\gamma_1^h,\dots,\gamma_h^h$ in
$C^2[0,T]$ such that $v_h(t):=\sum_{j=1}^h\gamma_j^h(t)w_j$ solves, for any $\eta\in W_h$, 
the problem
\begin{equation}\label{eq4}
\begin{gathered}
\int_\Omega\Big[i\dot v_h(t)+\Delta v_h(t)-\alpha v_h+\omega \Delta\dot v_h(t)
-\mu\dot v_h(t)-f(u)\Big]\eta=0, \\
v_h(0)=u_0^h.
\end{gathered}
\end{equation}
Taking $\eta=\bar w_j$ in \eqref{eq4}, we obtain
\begin{gather*}
(-i+\omega\lambda_j+\mu)\dot\gamma_j^h(t)+(\alpha+\lambda_j)\gamma_j^h(t)
=-\int_\Omega f(u(t))\bar w_j\,dx, \\
\gamma_j^h(0)=\lambda_j\Re\Big(\int_\Omega\bar u_0w_j\,dx\Big).
\end{gather*}
Since $\int_\Omega f(u(t))w_j\,dx\in C[0,T]$, we have a unique solution
 $\gamma_j^h$ to the previous problem. This yields to a solution $v_h$ 
defined as above and satisfying \eqref{eq4}. 
In particular, $v_h\in C^2([0,T],H_0^1)$. Taking $\eta=\overline{\dot v_h}$ 
in \eqref{eq4}, yields
\begin{align*}
&\|\nabla v_h(t)\|_{L^2}^2+\alpha\|v_h(t)\|_{L^2}^2+2\int_0^t\|\dot v_h(s)\|_*^2\,ds\\
&=\|\nabla u^h_0\|_{L^2}^2+\alpha\|u^h_0\|_{L^2}^2
 -2\int_0^t\Re(\int_\Omega\overline{f(u(s))}{\dot{v}_h}(s)\,dx)\,ds.
\end{align*}
Now, by Moser-Trudinger inequality, via the identity 
$2|ab|\leq \delta |a|^2+\frac1\delta |b|^2$, for $\delta>0$ near to zero, we have
\begin{align*}
2\int_0^t\Re\Big(\int_\Omega\overline{ f(u(s))}\dot v_h(s)\,dx\Big)\,ds
&\leq \frac1\delta\int_0^t\int_\Omega |f(u(s))|^2\,dx\,ds
 +\delta\int_0^t\int_\Omega|\dot v_h(s)|^2\,dx\,ds\\
&\leq \frac1\delta C_T+\delta\int_0^t\int_\Omega|\dot v_h(s)|^2\,dx\,ds\\
&\leq C_T+\int_0^t\|\dot v_h(s)\|_*^2\,ds.
\end{align*}
In fact, with Moser-Trudinger inequality, for any $0<\alpha<\frac{4\pi}{\|u\|_T^2}$,
\begin{align*}
\int_\Omega |f(u(s))|^2\,dx
&\leq C_\alpha\int_\Omega\Big(e^{\alpha\|u\|_T^2 
 (\frac{|u(s)|}{\|u\|_{T}})^2}-1\Big)\,dx\\
&\leq C_\alpha\int_\Omega |u(s)|^2\,dx\leq C_\alpha\|u\|_T^2=C_T.
\end{align*}
Thus, $\| v_h\|_{T}^2+\int_0^T\|\dot v_h(t)\|_*^2\leq C_T$.
So, $\{v_h\}$ is bounded in $H_0^1((0,T)\times\Omega)$. 
Then, taking the weak limit $v_h \rightharpoonup v$ in \eqref{eq4}, we obtain
a weak solution $v$ to \eqref{eq2}. 
Since $v\in H_0^1((0,T)\times\Omega)$, we obtain $v\in C([0,T], H_0^1(\Omega))$.
The existence part of the Lemma is proved.

 Now, for two solutions $v_1,v_2$ of \eqref{eq2} and $w:=v_1-v_2$, 
subtracting the equations and testing with $\overline{\dot w}$, we obtain
$$
\|\nabla w(t)\|_{L^2}^2+\alpha\| w(t)\|_{L^2}^2+2\int_0^t\|w(s)\|_*^2\,ds=0.
$$
The proof of Lemma \ref{lem1} is complete.
\end{proof}

We are ready to prove local well-posedness of \eqref{eq1}.
We denote $R_0:=\|\nabla u_0\|_{L^2}$, and for $R>0$ define the
closed subset of the complete metric space $E_T$,
$$
X_T:=\{u\in E_T: \|u\|_T\leq R,\; u(0)=u_0\}.
$$
Take the function $\phi(u):=v$, the solution to \eqref{eq2}. 
We shall prove that, for some $T,R>0$, $\phi$ is a contraction on $X_T$. 
Recall the identity
\begin{align*}
&\|\nabla v(t)\|_{L^2}^2+\alpha\|v(t)\|_{L^2}^2+2\int_0^t\|\dot v(s)\|_*^2\,ds\\
&=\|\nabla u_0\|_{L^2}^2+\alpha\|u_0\|_{L^2}^2-2\int_0^t
\Re\Big(\int_\Omega \overline{f(u(s))}\dot v(s)\,dx\Big)\,ds.
\end{align*}
Moreover, for any $0<\delta <\min\{\mu,4\pi/R^2\}$, by
 Moser-Trudinger inequality
\begin{align*}
&2\int_0^t\int_\Omega |f(u(s))||\dot v(s)|\,ds\\
&\leq \frac1\delta\int_0^t\int_\Omega |f(u(s))|^2\,ds
 +\delta\int_0^t\int_\Omega\dot |v(s)|^2\,ds\\
&\leq \frac{C_\delta}\delta \int_0^t\int_\Omega \Big(e^{\delta |u|^2}-1\Big)\,ds
 +\delta\int_0^t\int_\Omega|\dot v(s)|^2\,ds\\
&\leq \frac{C_\delta}\delta \int_0^t\int_\Omega |u(s)|^2\,ds
 +\delta\int_0^t\int_\Omega|\dot v(s)|^2\,ds\\
&\leq \frac{C_\delta}\delta TR^2+\frac\delta\mu\int_0^t\|\dot v(s)\|_*^2\,ds
 \leq\frac{C_\delta}\delta TR^2+\int_0^t\|\dot v_h(s)\|_*^2\,ds.
\end{align*}
This implies
$$
\|\nabla v(t)\|_{L^2}^2+\alpha\|v(t)\|_{L^2}^2\leq C\alpha R_0^2
+\frac{C_\delta}\delta TR^2.
$$
Taking $R^2>2C\alpha R_0^2$, yields
$$
\|v\|_T^2\leq \Big(\frac12+\frac{C_\delta}\delta T\Big)R^2.
$$
So $\phi(X_T)\subset X_T$ for small $T>0$. Let prove that $\phi$ is contractive. 
Take $u_1,u_2\in X_T$,  $v_i:=\phi(u_i)$, $v=v_1-v_2$ and $u=u_1-u_2$. 
Then, for any $\eta\in H_0^1$ and almost every $t\in[0,T]$,
$$
\int_\Omega\Big(i\dot v\eta-\alpha v\eta-\nabla v\nabla\eta
-\omega\nabla\dot v\nabla\eta-\mu\dot v\eta\Big)\,dx
=\int_\Omega\Big( f(u_1)-f(u_2)\Big)\eta\,dx.
$$
Taking the real part in the previous identity for $\eta=\overline{\dot v}$, 
via \eqref{f} yields, for any $\varepsilon>0$,
\begin{align*}
\|\nabla v(t)\|_{L^2}^2
&\leq \|\nabla v(t)\|_{L^2}^2+\alpha\|v(t)\|_{L^2}^2+2\int_0^t\|\dot v(s)\|_*^2\,ds\\
&= -2\int_0^t\Re\Big(\int_\Omega ( f(u_1)-f(u_2))\overline{\dot v}\,dx\Big)\,ds\\
&\leq \int_0^t\int_\Omega \Big[\frac1\varepsilon\Big| f(u_1)
 -f(u_2)\Big|^2+\varepsilon|\dot v|^2\Big]\,dx\,ds\\
&\leq \varepsilon\int_0^t\|\dot v\|_{L^2}^2\,ds
 +\frac1\varepsilon \int_0^t\int_\Omega \Big| f(u_1)-f(u_2)\Big|^2\,dx\,ds\\
&\leq \varepsilon\int_0^t\|\dot v\|_{L^2}^2\,ds
 +\frac1\varepsilon\int_0^t\Big[\int_\Omega|u|^2\Big(e^{\varepsilon |u_1|^2}-1
 +e^{\varepsilon |u_2|^2}-1\Big)\,dx\Big]\,ds.
\end{align*}
Now, for $0<\delta<\frac\pi{R^2}$, with Moser-Trudinger inequality via Sobolev 
embedding, we have
\begin{align*}
\int_\Omega |u|^2\Big(e^{\delta |u_1|^2}-1+e^{\delta |u_2|^2}-1\Big)\,dx
&\leq \|u\|_{L^4}^2\Big(\|e^{\delta |u_1|^2}-1\|_{L^2}
 +\|e^{\delta |u_2|^2}-1\|_{L^2}\Big)\\
&\leq \|u\|_{T}^2\Big(\|e^{2\delta |u_1|^2}-1\|_{L^1}^{\frac12}
 +\|e^{2\delta |u_2|^2}-1\|_{L^1}^{\frac12}\Big)\\
&\leq C_\delta\|u\|_{T}^2\Big(\| u_1\|_{L^2}
 +\| u_2\|_{L^2}\Big)\lesssim R\|u\|_{T}^2.
\end{align*}
Finally, taking $0<\varepsilon<\min\{2,\frac\pi{R^2}\}$, yields
$$
\|\phi(u_1)-\phi(u_2)\|_T\lesssim \sqrt{RT}\|u_1-u_2\|_T. 
$$
Thus $\phi$ is a contraction of $X_T$ for $T>0$ small enough. 
With Picard Theorem, there exists a unique fixed point $u$ which is a 
solution to \eqref{eq1}.
Uniqueness follows arguing as previously and applying the precedent inequality 
for two solutions to \eqref{eq1}, which belong to $X_T$ with a continuity 
argument for some $T>0$ small enough.


\subsection{Global existence in the defocusing case}

We recall two important facts. First, the time of local existence depends only on
the quantity $\|\nabla u_0\|_{L^2}$. Second the energy dominates the 
$H_0^1$ norm. Let $u$ be the maximal
solution of \eqref{eq1} in the space ${E}_T$ for any $0<T<{T^*}$ with initial 
data $u_0$,
where $0<T^*\leq +\infty$ is the lifespan of $u$. We shall prove that $u$ is global.
By contradiction, suppose that $T^*< +\infty$, we consider for $0<s<T^*$, 
the following problem
\begin{equation} \label{ePs}
\begin{gathered}
i\dot v+\Delta v-\alpha v+\omega\Delta\dot v-\mu\dot v = f(v),\\
v(s,.) = u(s,.),\\
v|_{\partial \Omega} = 0.
 \end{gathered}
\end{equation}
By the same arguments used in the local existence and taking
$$
0<\delta\leq\min\big\{\mu,\frac{\pi}{E(0)}\big\},
$$
we can find a real $\tau>0$ and a solution
$v$ to \eqref{ePs} on $[s,s+\tau]$.
According to the section of local existence and using decay of the energy,
$\tau$ does not depend on $s$. Thus, if we let $s$ be
close to $T^*$ such that $s+\tau>T^*$, we can extend $v$ for times
higher than $T^*$. This fact
contradicts the maximality of $T^*$. We obtain the result claimed
in Theorem \ref{thm1}.

\section{Proof of Theorem \ref{thm3}}

We are interested on the focusing case associated to the problem \eqref{eq1}, 
so here and hereafter, we fix $\epsilon=-1$. By \eqref{as} we have 
\cite{jc}, $m=J(\varphi)>0$ where $\varphi$ is the ground state solution of
$$
-\Delta\varphi+\alpha\varphi=f(\varphi).
$$
If there exists $t_0>0$ such that $u(t_0)\notin N^+$, then 
$I(t_0)\leq 0$. With a continuity argument, there exists a time $t_1\in(0,t_0)$ 
such that $I(t_1)=0$ and $E(t_1)<m$ which contradicts the definition of $m$.
Let us prove that $u$ is global. For any real number $0<\varepsilon<1$,
\begin{align*}
E(t)
&=\alpha\|u\|_{L^2}^2+\|\nabla u\|_{L^2}^2-\int_\Omega F(|u|^2)\,dx\\
&=\alpha\|u\|_{L^2}^2+\varepsilon\|\nabla u\|_{L^2}^2+(1-\varepsilon)
 \Big(I(t)-\alpha\|u\|_{L^2}^2+\int_\Omega \bar uf(u)\,dx\Big)\\
&\quad -\int_\Omega F(|u|^2)\,dx\\
&\geq \alpha\varepsilon\|u\|_{L^2}^2+\varepsilon\|\nabla u\|_{L^2}^2
 +(1-\varepsilon)I(t)+((1-\varepsilon)(1+a)-1)\int_\Omega F(|u|^2)\,dx.
\end{align*}
Thus, using the fact that $f$ satisfies \eqref{as}, we have for any 
$0<\varepsilon<\frac a{2+a}$,
\begin{equation}\label{e}
E(0)\geq E(t)\geq \alpha\varepsilon\|u\|_{L^2}^2
+\varepsilon\|\nabla u\|_{L^2}^2+\varepsilon\int_\Omega F(|u|^2)\,dx.
\end{equation}
Thus $\|\nabla u(t)\|_{L^2}$ is bounded and $u$ is global.

Now, we prove an exponential decay of the solution to \eqref{eq1}.
Note that since $u(t)\in N^+$ and $f$ satisfies \eqref{as}, we have 
$E\geq I>0$. We denote, for some $0<\varepsilon<\min\{\alpha,\frac a{2+a}\}$ 
(so satisfying \eqref{e}), the real function
$$
L(t):=E(t)+\frac{\varepsilon\omega}2\int_\Omega|\nabla u(t)|^2\,dx.
$$
By \eqref{e}, we have $E\lesssim L\lesssim E$.
Taking account of \eqref{eq1}, we compute, for $0<\varepsilon<\frac\mu{1+\mu}$,
\begin{align*}
\dot L
&= \dot E-\varepsilon\Big(\alpha\|u\|_{L^2}^2+\|\nabla u\|_{L^2}^2
 -\int_\Omega \bar uf(u)\,dx+\int_\Omega[\mu \Re(\bar u\dot u)
 -\Im(\bar u\dot u)]\,dx\Big)\\
&\leq -\mu\|\dot u\|_{L^2}^2-\varepsilon\Big(E+\int_\Omega F(|u|^2)\,dx
 -\int_\Omega \bar uf(u)\,dx+\int_\Omega[\mu \Re(\bar u\dot u)
 -\Im(\bar u\dot u)]\,dx\Big)\\
&\leq -\mu\|\dot u\|_{L^2}^2-\varepsilon E+\varepsilon\int_\Omega|uf(u)|\,dx
 +\frac\varepsilon2(1+\mu)(\|\dot u\|_{L^2}^2+\|u\|_{L^2}^2)\\
&\leq -\varepsilon E+\varepsilon\int_\Omega|uf(u)|\,dx
 +\frac\varepsilon2(1+\mu)\|u\|_{L^2}^2.
\end{align*}
With \eqref{e}, we have $E\geq \varepsilon \|u\|_{H_0^1}^2$, thus,
 using Moser-Trudinger inequality, for any $0<\delta<\frac{4\pi\varepsilon}{E(0)}$,
\begin{align*}
\int_\Omega |uf(u)|\,dx
&\leq {C_\varepsilon}\int_\Omega \Big(e^{\delta |u|^2}-1\Big)\,dx\\
&\leq {C_\varepsilon}\int_\Omega \Big(e^{\delta\|u\|_{H_0^1}^2 (\frac {|u|}{\|u\|_{H_0^1}})^2}-1\Big)\,dx\\
&\leq {C_\varepsilon}\|u\|_{L^2}^2.
\end{align*}
So, 
\[
\dot L(t)
\leq -\varepsilon \Big(E-(C_\varepsilon+\frac{1+\mu}2)\|u\|_{L^2}^2\Big).
\]
Now, also with \eqref{e}, for 
$\alpha>\frac2\varepsilon[\frac{1+\mu}2+C_\varepsilon]$, 
we have 
$$
E-\big(\frac{1+\mu}2+C_\varepsilon\big)\|u\|_{L^2}^2\geq \frac\varepsilon2 E.
$$
Finally, we conclude with a Gronwall argument via the inequalities
$$
\dot L(t)\lesssim -E(t)\lesssim -L(t).
$$


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\end{document}